Recent developments in bottom-up synthesis of atomically precise graphene nanoribbons (GNRs) are paving the way for new technologies with angstrom-scale features that host a myriad of electronic, spin, and topological properties. Studying the ultrafast phenomena relevant to the optoelectronic potential of these GNRs requires extreme spatio-temporal resolution. Conventional techniques for atomic-scale investigation of materials, such as scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS), lack the necessary time resolution to investigate ultrafast dynamics. Meanwhile, conventional ultrafast measurements are able to time-resolve the optical excitations of these GNRs, but, they lack the requisite spatial resolution to perform more precise experiments on isolated individual structures. Instead, they act on ensembles and average over many GNR lengths, orientations, and sample qualities.With the development of terahertz scanning tunneling microscopy (THz-STM), coherent control of tunneling electrons on sub-picosecond timescales is now possible in an STM tunnel junction. However, to fully empower THz-STM as a scientific tool for materials science, an approach to characterize the electronic properties of new materials and nanostructures without prior knowledge of the sample response is needed. This demands the establishment of terahertz scanning tunneling spectroscopy (THz-STS) with atomic spatial resolution and an approach to analyzing such measurements.In this thesis, we demonstrate atomic-scale THz-STS for the first time by extracting the differential conductance (dI/dV) from a model GNR system with angstrom-scale resolution in all three spatial dimensions. We utilize bottom-up synthesis to construct atomically precise 7-atom-wide armchair graphene nanoribbons (7-AGNR) on an Au(111) substrate. THz-STM is applied to probe the electronic structure of a 7-AGNR while operating at ultralow tip heights, where the distance between STM tip and GNR is reduced by ~3.0-4.0 A. compared to conventional STM tip heights. We construct a model of the GNR electronic density of states and apply THz-STS to a collection of THz-STM images taken at different incident THz field strengths to construct dI/dV maps of the frontier orbitals on a 7-AGNR segment. Lastly, a set of time-domain measurements are performed in the hopes of observing dynamical effects associated with charging of the GNR during charge injection by the THz-STM. The aforementioned steady-state THz-STS results are sufficient to describe the time-domain measurements without charging, suggesting further sample preparation is needed to decouple the GNR from the Au(111) surface and reveal transient effects.We further develop and test an algorithm, based on a polynomial representation of the junction I-V curve, for extracting the tunnel junction dI/dV accessed by an ultrafast THz-STM probe in both steady-state and optical-pump/THz-STS-probe experiments. We test our algorithm against a set of model systems constructed to represent typical STM samples, e.g., metals, semiconductors, and molecules. Then, we simulate a THz-STS experiment wherein we address systematic error by defining a procedure to determine the algorithm parameters. After setting the stage with steady-state THz-STS, our algorithm is expanded upon, and we show how a dynamic junction I(V) predicted for an optical-pump/THz-STM-probe experiment can be extracted from THz-STS measurements. By applying the cross-correlation theorem in the frequency domain we find that the time-resolution of THz-STS is not limited to the input bandwidth of the ultrafast THz voltage pulse. As a result, our pump-probe THz-STS algorithm reveals dynamics faster than a single oscillation cycle of the THz transient,and in the future may be combined with atomically resolved THz-STS experiments to bridge a new frontier of ultrafast materials science.
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